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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj873 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33745. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj873.4 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj873.7 | ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) |
bnj873.8 | ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) |
Ref | Expression |
---|---|
bnj873 | ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj873.4 | . 2 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
2 | nfv 1917 | . . 3 ⊢ Ⅎ𝑔∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
3 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑓𝐷 | |
4 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑓 𝑔 Fn 𝑛 | |
5 | bnj873.7 | . . . . . 6 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) | |
6 | nfsbc1v 3777 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜑 | |
7 | 5, 6 | nfxfr 1855 | . . . . 5 ⊢ Ⅎ𝑓𝜑′ |
8 | bnj873.8 | . . . . . 6 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) | |
9 | nfsbc1v 3777 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜓 | |
10 | 8, 9 | nfxfr 1855 | . . . . 5 ⊢ Ⅎ𝑓𝜓′ |
11 | 4, 7, 10 | nf3an 1904 | . . . 4 ⊢ Ⅎ𝑓(𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
12 | 3, 11 | nfrexw 3307 | . . 3 ⊢ Ⅎ𝑓∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
13 | fneq1 6613 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)) | |
14 | sbceq1a 3768 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ [𝑔 / 𝑓]𝜑)) | |
15 | 14, 5 | bitr4di 288 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ 𝜑′)) |
16 | sbceq1a 3768 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ [𝑔 / 𝑓]𝜓)) | |
17 | 16, 8 | bitr4di 288 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ 𝜓′)) |
18 | 13, 15, 17 | 3anbi123d 1436 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
19 | 18 | rexbidv 3177 | . . 3 ⊢ (𝑓 = 𝑔 → (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
20 | 2, 12, 19 | cbvabw 2805 | . 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
21 | 1, 20 | eqtri 2759 | 1 ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1087 = wceq 1541 {cab 2708 ∃wrex 3069 [wsbc 3757 Fn wfn 6511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-sbc 3758 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-br 5126 df-opab 5188 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-fun 6518 df-fn 6519 |
This theorem is referenced by: bnj849 33660 bnj893 33663 |
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